Claude Dutheillet

On Well-Formed Coloured Nets and Their Symbolic Reachability Graph

The new class of Well Formed Coloured Nets (WN) is formally defined as an extension of Regular Nets (RN), together with an extended Symbolic Reachability Graph (SRG) construction algorithm. WNs allow the representation of any colour function in a structured form, so that they have the same modelling power as general coloured nets (CPN). In particular, with respect to RN, WNs allow the use of non-symmetric initial markings, of repeated occurrences of the same basic class in the Cartesian product defining the colours for transitions and places, and of the “constant” and “successor” functions as arc labels. The SRG allows colour symmetries to be exploited to reduce the space and time complexity of the analysis by reachability graph. The advantage of using WNs instead of unconstrained CPNs is that the detection of symmetries to construct the SRG is totally algorithmic, and requires no special heuristics.

A symbolic reachability graph for coloured Petri nets

Coloured Petri nets are well suited to the modelling of symmetric systems. Model symmetries can be usefully exploited for the sake of analysis efficiency as well as for modelling convenience. We present a reduced reachability graph called symbolic reachability graph that enjoys the following properties: (1) it can be constructed directly by an efficient algorithm without considering the actual state space of the model; (2) it can be substantially smaller than the ordinary reachability graph; (3) its analysis provides equivalent results as the analysis of the ordinary reachability graph. The construction procedure for the symbolic reachability graph is completely effective in the case of a syntactically restricted class of coloured nets called “well-formed nets”, while for the unrestricted case of coloured nets some procedures may not be easily implementable in algorithmic form.

On Liveness in Extended non Self-Controlling Nets

For several years, research has been done to establish relations between the liveness of a net and the structure of the underlying graph. This work has resulted in the proposition of polynomial algorithms to check liveness for particular classes of nets. In this paper, we present Extended Non Self-Controlling Nets, a class of nets that includes Extended Free-Choice Nets and Non Self-Controlling Nets. We develop some properties of this new class of nets and we propose polynomial algorithms whose application domain is wider than the domain of the previous algorithms.

Regular stochastic Petri nets

An extension of regular nets, a class of colored nets, to a stochastic model is proposed. We show that the symmetries in this class of nets make it possible to develop a performance evaluation by constructing only a graph of symbolic markings, which vertices are classes of states, instead of the whole reachability graph. Using algebraic techniques, we prove that all the states in a class have the same probability, and that the coefficients of the linear system describing the lumped Markov process can be calculated directly from the graph of symbolic markings.

Stochastic Well-Formed Coloured Nets and Multiprocessor Modelling Applications

The new class of Stochastic Well-formed coloured Nets (SWN) is defined as a syntactic restriction of Stochastic High-Level Nets. The interest of the introduction of restrictions in the model definition is the possibility of exploiting the Symbolic Reachability Graph (SRG) to reduce the complexity of Markovian performance evaluation with respect to classical Petri net techniques. It turns out that SWNs allow the representation of any colour function in a structured form, so that any unconstraint high-level net can be transformed into a well formed net. Moreover, most constructs useful for the modelling of distributed computer systems and architectures directly match the “well form” restriction, without any need of transformation. A non trivial example of the usefulness of the technique in the performance modelling and evaluation of multiprocessor architectures is included.

An algorithmic approach for analysis of finite-source retrial systems with unreliable servers

This paper aims at presenting an approach for analyzing finite-source retrial systems with servers subject to breakdowns and repairs, using Generalized Stochastic Petri Nets (GSPNs). This high-level formalism allows a simple representation of such systems with different breakdown disciplines. From the GSPN model, a Continuous Time Markov Chain (CTMC) can be automatically derived. However, for multiserver retrial systems with unreliable servers, the models may have a huge state space. Using the GSPN model as a support, we propose an algorithm for directly computing the infinitesimal generator of the CTMC without generating the reachability graph. In addition, we develop the formulas of the main stationary performance and reliability indices, as a function of the number of servers, the size of the customer source and the stationary probabilities. Through numerical examples, we discuss the effect of the system parameters and the breakdown disciplines on performance.

Colored stochastic Petri nets for modelling and analysis of multiclass retrial systems

Most retrial models assume that customers and servers are homogeneous. However, multiclass (or heterogeneous) retrial systems arise in various practical areas such as telecommunications and cellular mobile networks. Multiclass models are far more difficult for mathematical analysis than single class ones. So, explicit results are available only in few special cases. Actually, so far multiclass retrial systems have been analyzed only by means of queueing theory and almost all studies consider models with several customers classes and a service station consisting in one single server or multiple homogeneous (identic) servers and an infinite population size. In this paper, we propose an approach for modelling and analyzing finite-source retrial systems with several customers classes and servers classes using the Colored Generalized Stochastic Petri Nets (CGSPNs). This high-level mathematical model is appropriate for describing and analyzing the performance of systems exhibiting concurrency and synchronization, possibly with heterogeneous components. Using a high-level formalism makes the description of the system easier, while preserving the possibility of obtaining exact performance results. We show how the main steady-state performance indices can be derived and we analyze the behaviour of heterogeneous retrial systems under two service disciplines. The numerical results are graphically displayed to illustrate the effect of system parameters and service discipline on the mean response time.

Exploiting Partial Symmetries for Markov Chain Aggregation

Abstract The technique presented in this paper allows the automatic construction of a lumped Markov chain for almost symmetrical Stochastic Well-formed Net (SWN) models. The starting point is the Extended Symbolic Reachability Graph (ESRG), which is a reduced representation of a SWN model reachability graph (RG), based on the aggregation of states into classes. These classes may be used as aggregates for lumping the Continuous Time Markov Chain (CTMC) isomorphic to the model RG: however it is not always true that the lumpability condition is verified by this partition of states. In the paper we propose an algorithm that progressively refines the ESRG classes until a lumped Markov chain is obtained.

An efficient algorithm for finding structural deadlocks in colored Petri nets

In this paper, we present an algorithm to compute structural deadlocks in colored nets under specified conditions. Instead of applying the ordinary algorithm on the unfolded Petri net, our algorithm takes advantage of the structure of the color functions. It is obtained by iterative optimizations of the ordinary algorithm. Each optimization is specified by a meta-rule, whose application is detected during the computation of the algorithm. The application of such meta-rules speeds up a step of the algorithm with a factor proportional to the size of a color domain. We illustrate the efficiency of this algorithm compared to the classical approach on a colored net modelling the dining philosophers problem.

On the use of exact lumpability in partially symmetrical well-formed nets

Well-formed nets (WNs) have proved an efficient model for building quotient reachability graphs that can be used either for qualitative or performance analysis. However, local asymmetries often break any possibility of grouping states into classes, thus drastically reducing the interest of the approach. An efficient solution has been proposed for qualitative analysis, which relies on a separate representation of the asymmetries in a so called control automaton. The quotient graph is then obtained by synchronizing the transitions of the WN model with the transitions of the control automaton. In this paper, we improve this approach to quantitative analysis. We show that it can be used to build an aggregated graph that is isomorphic to a Markov chain which verifies exact lumpability. Theoretical considerations and practical experiments show that our method outperforms previous approaches.